Section 55: Problem 2 Solution
Working problems is a crucial part of learning mathematics. No one can learn topology merely by poring over the definitions, theorems, and examples that are worked out in the text. One must work part of it out for oneself. To provide that opportunity is the purpose of the exercises.
James R. Munkres
Show that if
is nulhomotopic, then
has a fixed point and
maps some point
to its antipode
.
If
is nulhomotopic, then it is continuous and extendable to a continuous function
(Lemma 55.3),
has a fixed point
(Theorem 55.6) in
, so that
. Similarly, if
is the rotation of
by 180 degrees, then
is nulhomotopic, and for some point
.